3.104 \(\int \frac{\sqrt{c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=540 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C d f-a b (3 B d f+c C f+C d e)+b^2 (3 A d f+c C e)\right )}{3 b^2 f (b c-a d) (b e-a f)}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (8 a^2 C d f^2-a b f (6 B d f+c C f+3 C d e)+b^2 (3 d f (A f+B e)-C e (2 d e-c f))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f^2 \sqrt{c+d x} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)}+\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (4 a C f-3 b B f+2 b C e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f^2 \sqrt{c+d x} \sqrt{e+f x}} \]

[Out]

(2*(4*a^2*C*d*f + b^2*(c*C*e + 3*A*d*f) - a*b*(C*d*e + c*C*f + 3*B*d*f))*Sqrt[a
+ b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b^2*(b*c - a*d)*f*(b*e - a*f)) - (2*(A*b^
2 - a*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a*f)*Sqr
t[a + b*x]) + (2*Sqrt[-(b*c) + a*d]*(8*a^2*C*d*f^2 - a*b*f*(3*C*d*e + c*C*f + 6*
B*d*f) + b^2*(3*d*f*(B*e + A*f) - C*e*(2*d*e - c*f)))*Sqrt[(b*(c + d*x))/(b*c -
a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]]
, ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^3*Sqrt[d]*f^2*(b*e - a*f)*Sqrt[c + d*x]
*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(d*e - c*f)*(2*b*C*e -
3*b*B*f + 4*a*C*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*
f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f
)/(d*(b*e - a*f))])/(3*b^3*Sqrt[d]*f^2*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi [A]  time = 3.08539, antiderivative size = 540, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C d f-a b (3 B d f+c C f+C d e)+b^2 (3 A d f+c C e)\right )}{3 b^2 f (b c-a d) (b e-a f)}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (8 a^2 C d f^2-a b f (6 B d f+c C f+3 C d e)+b^2 (3 d f (A f+B e)-C e (2 d e-c f))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f^2 \sqrt{c+d x} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)}+\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (4 a C f-3 b B f+2 b C e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f^2 \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(3/2)*Sqrt[e + f*x]),x]

[Out]

(2*(4*a^2*C*d*f + b^2*(c*C*e + 3*A*d*f) - a*b*(C*d*e + c*C*f + 3*B*d*f))*Sqrt[a
+ b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b^2*(b*c - a*d)*f*(b*e - a*f)) - (2*(A*b^
2 - a*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a*f)*Sqr
t[a + b*x]) + (2*Sqrt[-(b*c) + a*d]*(8*a^2*C*d*f^2 - a*b*f*(3*C*d*e + c*C*f + 6*
B*d*f) + b^2*(3*d*f*(B*e + A*f) - C*e*(2*d*e - c*f)))*Sqrt[(b*(c + d*x))/(b*c -
a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]]
, ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^3*Sqrt[d]*f^2*(b*e - a*f)*Sqrt[c + d*x]
*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(d*e - c*f)*(2*b*C*e -
3*b*B*f + 4*a*C*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*
f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f
)/(d*(b*e - a*f))])/(3*b^3*Sqrt[d]*f^2*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(3/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 12.2866, size = 551, normalized size = 1.02 \[ -\frac{2 \left (b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left (-8 a^2 C d f^2+a b f (6 B d f+c C f+3 C d e)+b^2 (C e (2 d e-c f)-3 d f (A f+B e))\right )-i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (8 a^2 C d f^2-a b f (6 B d f+c C f+3 C d e)+b^2 (3 d f (A f+B e)+C e (c f-2 d e))\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )-i b f (a+b x)^{3/2} (d e-c f) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (4 a^2 C d f-a b (3 B d f+c C f+C d e)+b^2 (3 A d f+c C e)\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )+b^2 d f (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left (3 f \left (a (a C-b B)+A b^2\right )-C (a+b x) (b e-a f)\right )\right )}{3 b^4 d f^2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(3/2)*Sqrt[e + f*x]),x]

[Out]

(-2*(b^2*Sqrt[-a + (b*c)/d]*(-8*a^2*C*d*f^2 + a*b*f*(3*C*d*e + c*C*f + 6*B*d*f)
+ b^2*(-3*d*f*(B*e + A*f) + C*e*(2*d*e - c*f)))*(c + d*x)*(e + f*x) + b^2*Sqrt[-
a + (b*c)/d]*d*f*(c + d*x)*(e + f*x)*(3*(A*b^2 + a*(-(b*B) + a*C))*f - C*(b*e -
a*f)*(a + b*x)) - I*(b*c - a*d)*f*(8*a^2*C*d*f^2 - a*b*f*(3*C*d*e + c*C*f + 6*B*
d*f) + b^2*(3*d*f*(B*e + A*f) + C*e*(-2*d*e + c*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c
+ d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqr
t[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*b*f*(d*e -
c*f)*(4*a^2*C*d*f + b^2*(c*C*e + 3*A*d*f) - a*b*(C*d*e + c*C*f + 3*B*d*f))*(a +
b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*E
llipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a
*d*f)]))/(3*b^4*Sqrt[-a + (b*c)/d]*d*f^2*(b*e - a*f)*Sqrt[a + b*x]*Sqrt[c + d*x]
*Sqrt[e + f*x])

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Maple [B]  time = 0.059, size = 4732, normalized size = 8.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(3/2)/(f*x+e)^(1/2),x)

[Out]

2/3*(3*A*x*b^4*d^2*e*f^2-3*B*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/
d/(a*f-b*e))^(1/2))*a^2*b^2*d^2*e*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a
*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+3*B*EllipticF(((b*x+a)*d/(a*d-b*c))^
(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*d^2*e^2*f*((b*x+a)*d/(a*d-b*c))^(1/
2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*B*EllipticF(((b*x
+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c*d*e^2*f*((b*x+a)*d
/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-9*C*
EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*c*d
*f^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b
*c))^(1/2)-11*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^
(1/2))*a^3*b*d^2*e*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*
(-(d*x+c)*b/(a*d-b*c))^(1/2)+C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*
f/d/(a*f-b*e))^(1/2))*a^2*b^2*d^2*e^2*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/
(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*C*EllipticE(((b*x+a)*d/(a*d-b*c)
)^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c^2*e*f^2*((b*x+a)*d/(a*d-b*c))^(
1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-4*C*EllipticF(((b
*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*c*d*f^3*((b*x+a)
*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+4*
C*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*d
^2*e*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a
*d-b*c))^(1/2)-2*C*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e
))^(1/2))*a^2*b^2*d^2*e^2*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(
1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*C*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*
d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c^2*e*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+
e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*A*EllipticE(((b*x+a)*d/(a*d
-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c*d*f^3*((b*x+a)*d/(a*d-b*c)
)^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*A*EllipticE(
((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*d^2*e*f^2*((b
*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/
2)+3*A*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^
4*c*d*e*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b
/(a*d-b*c))^(1/2)+6*B*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-
b*e))^(1/2))*a^2*b^2*c*d*f^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^
(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+9*B*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a
*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*d^2*e*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f
*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*B*a*b^3*c*d*e*f^2+4*C*a^
2*b^2*c*d*e*f^2-C*a*b^3*c*d*e^2*f+3*A*x^2*b^4*d^2*f^3-9*B*EllipticE(((b*x+a)*d/(
a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c*d*e*f^2*((b*x+a)*d/(a*d
-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+13*C*Elli
pticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c*d*e
*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b
*c))^(1/2)-2*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(
1/2))*a*b^3*c*d*e^2*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(
-(d*x+c)*b/(a*d-b*c))^(1/2)-2*C*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)
*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c*d*e*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b
/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+4*C*EllipticF(((b*x+a)*d/(a*d-b*c
))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c*d*e^2*f*((b*x+a)*d/(a*d-b*c))^
(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+8*C*EllipticE(((
b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^4*d^2*f^3*((b*x+a)*
d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*B
*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*d^
2*e^2*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*
d-b*c))^(1/2)+3*B*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e)
)^(1/2))*b^4*c*d*e^2*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*
(-(d*x+c)*b/(a*d-b*c))^(1/2)+3*B*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c
)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c*d*f^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/
(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*C*EllipticF(((b*x+a)*d/(a*d-b*c)
)^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*d^2*e^3*((b*x+a)*d/(a*d-b*c))^(1/
2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*C*EllipticF(((b*x
+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c^2*e^2*f*((b*x+a)*d
/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+2*C*
EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c*d*e
^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c
))^(1/2)+3*A*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/
2))*a^2*b^2*d^2*f^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(
d*x+c)*b/(a*d-b*c))^(1/2)+C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d
/(a*f-b*e))^(1/2))*a^2*b^2*c^2*f^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-
b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*B*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/
2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c^2*f^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-
(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+3*B*EllipticF(((b*x+a)*d
/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c^2*e*f^2*((b*x+a)*d/(a*d
-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-6*B*Ellip
ticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*d^2*f^3*
((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^
(1/2)+2*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))
*a*b^3*d^2*e^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c
)*b/(a*d-b*c))^(1/2)+C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f
-b*e))^(1/2))*b^4*c^2*e^2*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(
1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*
d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c*d*e^3*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b
/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+4*C*EllipticF(((b*x+a)*d/(a*d-b*c
))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c^2*f^3*((b*x+a)*d/(a*d-b*c))^
(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*B*x^2*a*b^3*d^
2*f^3+4*C*x^2*a^2*b^2*d^2*f^3-C*x^2*b^4*d^2*e^2*f+3*A*x*b^4*c*d*f^3+C*x^3*a*b^3*
d^2*f^3-C*x^3*b^4*d^2*e*f^2+3*A*b^4*c*d*e*f^2+C*x^2*a*b^3*c*d*f^3-C*x^2*b^4*c*d*
e*f^2-3*B*x*a*b^3*c*d*f^3-3*B*x*a*b^3*d^2*e*f^2+4*C*x*a^2*b^2*c*d*f^3+4*C*x*a^2*
b^2*d^2*e*f^2-C*x*a*b^3*d^2*e^2*f-C*x*b^4*c*d*e^2*f)*(f*x+e)^(1/2)*(b*x+a)^(1/2)
*(d*x+c)^(1/2)/d/b^4/f^2/(a*f-b*e)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*
f*x+a*d*e*x+b*c*e*x+a*c*e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{f x + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(3/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)), x)